Collatz-problem;
approximation 2S
to 3N
a large scale view; N = 1..1038
; a hullcurve N=1…10250
and some more data up to N~102400

1.A
large-scale view into the distance of 3N and the next perfect power of 2

1.1.The problem

In the
investigation of the possibility of 1-cycles under the Collatz-iteration (see [1]), we find, that the length N of
such a hypothetic cycle and the distance of 3N to the next greater power 2S
is a crucial relation. Here S is defined such that 2S > 3N
and more precisely, such that

S = ceil (N log(3)/log(2) )

which, in
relation to the collatz-problem means: N is the number of operations 3x+1,
and S
is the number of x/2-operations
if we consider the 1-cycle (or "primitive-loop" in my older
formulation). In the notation of Ray Steiner a "1-cycle" beginning at its
least element a
is a sequence of length N of (3x+1)/2 operations followed by A-1x/2
operations such that N+A-1=S in my notation. Although that problem of
the existence of an 1-cycle was solved to the negative by Ray Steiner
already in 1977 using a result of G. Rhin, the bound, which occurs there seems
very rough. So to get an own impression of the behaviour of the approximability
of 2S
to 3N
(where N
is given) I produced data using the arbitrary-precision software Pari/GP.

We are
interested in the relative value of that distance (2S-3N)/3N=2S/3N-1,
and use the log,
because the values of the original function cannot be used with a computer if N itself
goes to hundreds or thousand digits.

1.2.The function f(N) under consideration and the
hullcurve of moving minima

The values f(N)
range from 0
to log(2)~0.69314
(exclusive the bounds); a "good" approximation (small relative
distance) is f(N)
near zero, a "bad" approximation is f(N) near log(2). Also "good"
approximations improve with increasing N, but we don't have an easy relation of N to
that measure of quality of approximation. So in plot 2 I show a trend, how it appears in
a large scale view for N=1 to 1038 .

1.3.First impression and the idea of a
hullcurve (or envelope)

But before the
plot with the trend another, more obvious plot to introduce into the idea of
envelope. Here I used N=1 to N=800 and the function-values for each N.

Plot 1: Plot
of approximants f(N) for some small N

We see a clear
regularity ; the apparent antidiagonals of slope ~-45°, for instance, occur, if we
increase N
in steps by 12
beginning at N=5.

However, if we
look at the points nearest the x-axis (y~0), then we see that there is also another
cyclicitiness overlaid: we have a very low point at N=41, then the next lower points
follow in steps of 53 up to N=306, where the rhythm breaks such that at N=306+53
we have a very high point instead.

This general
behave is simply due to modularity of f(N)mod(log(2)). However, it does not lead to exact
periodicity of the distances. The reason for this is the irrationality of the
ratio ß=log(3)/log(2)
and N.

Anyway, for the
analysis of the Collatz-problem, namely for the discussion of the "primitive loop" (or "1-cycle") we are interested
in the cases of best approximations. We may say, that cycles are "more likely" for length N, if
the distance-measure f(N) is small, or near zero. Well: "likeliness" is not really a correct
notion for the cycle-discussion, but it gives here an idea, that we are
interested in the sharpest approximations depending on N.

So we define a lower envelope for the f(N)-function
by connecting the moving minima of
the points when increasing N; in the above plot it is indicated by the blue
line.

1.4.The lower envelope for f(N)

We are then
interested in the characteristic of that envelope.

In the
following plot I draw that envelope, whose points were determined using the
continued-fraction-representation of log(3)/log(2). That representation gives the
points N
at which minimal values f(N) occur. That method allows to find relevant N up to N=1038
and its values f(N)
in a few seconds, so we get enough points for a good overview. Because we deal
with huge values of N and also very small values of f(N) I
used a logarithmic scale for both values. (For visual comparision I also inserted
sample points for some N which are not on the envelope of local minima)

Plot 2:

Surprisingly
the general tendency of the lower envelope envlow is linear in this log-scales.
This information is new to me; I didn't come across a linear ratio of that
logarithms before (in [1] I had some
logarithmic/polynomial guess bases on much less N).

The long-scale
relation between N
and envlow(f(N))
is then, using the above equation for the trend, in some different ways of expression:

Note: this does
not help for the general cycle. The trend
in this formulation is also very raw and rather useless for the actual computation
of the upperbound for the smallest member of the general cycle. Such
computation should be made on the base of the original computation for each N. In
table 2.2 I give some example data for cycle-lengthes N=1 to N=100 instead. That table shows the
lower bound for a1
to make a cycle impossible, or said differently: the smallest member of a
general cycle must be smaller than the given bound (but all of these small
values a are easily checked empirically and none is member of a cycle, so
cycles of the indicated lengthes cannot exist. For instance, if a1=5
is given, no cycle at all is possible – solely on the discussed approximation
argument).

1.5.More values and another hullcurve

I proceeded
using more terms for the hullcurve up to 1600 coordinates which means I arrived at N~10250.
Setting precision of computation in Pari/GP to 15000 gives about 14500
entries for the continued fraction of log(3)/log(2). Building the list of convergents
gives about 80000
values for N,
which gives the coordinates for the hullcurve.

To finally
compute with all these values I had then to reduce the precision to 5000
digits (because of memory management) and could effectively check the
coordinates up to N~ 102848 distributed in about 25000 entries in that table of
convergents. Because log(N) seems in the great overview nearly linear
with log(f(N))
I computed the difference of that values log10(N) – (-log10(f(N))) = log10(N*log(2^S/3^N))
. That differences were in a very near range of about –4 to +5 . However, also here we find
nonconstant bounds, so there is again the need of a greater overview: to look
at a hullcurve connecting that progressive extrema. Here is the Pari/GP-program and then the few coordinates
of that hullcurve.

cf_xopt_chklae( cfl ) \\ check length
when list of convergents were expanded to all N with diminuishing intervals \\ %2415 = 82099 \\
length of that list, this is too long for current memory allocation!

NList = cf_xopt( cfl , 25000); \\
create list containing all relevant N \\
allocated memory allows only to work with about 25000 entries \\ VE(NList,5) \\
do a short check , show the first few entries N \\ %2417 = [1, 3, 5, 17, 29]~ \\ the first few N (for the exponent of 3^N
according to the list of convergents

fmt(5000,12) \\
precision may be reduced, but must be >2000 (for instance) [lg10=log(10),lg2=log(2),lg3=log(3),ld3=lg3/lg2] \\ recompute constants with current prec

Note: (*1) (21.Mar) I computed the
new values by a shorter routine and another list, so the list-index was not
comparable

According to
this, up to N~1010853
we have

log(2S/3N)
> 1/(N*104.07 )

where it is not
yet recognizable, which form (linear, parabolic, logarithmic,…) that hullcurve
shows in a even longer scale. (If we use the log10 of log10(N)
again, it looks as if we have again a bound roughly linear with that values…)

Gottfried
Helms, 3'2012 (previous version: 3'2010)

for a overview
of the related cycle-discussion ("primitive loop") see:

2.2.Table 2: upper bound for the minimal
member "a"
of an assumed general cycle of length N (also based on the assumtion of maximal density
of set of members)

"Maximal density
of members" means, the elements (a1,a2,… an)=(5,7,11,13,…,an)
or (a1,a2,…
an)=(7,11,13,…,an) or similarly, generally of
the form ak+2
= ak+6 and ak+1=ak+2 or ak+1 = ak+4.
To make a cycle possible at all, its minimal member a1 must be smaller than
the upper bound a1
given in the table. The higher the upper bound, the "easier" it is for
a cycle to exist:

N (cycle length)

upper bound a1

N (cycle length)

upper bound a1

1

5

51

83

2

5

52

5

3

5

53

5

4

5

54

17

5

25

55

5

6

5

56

47

7

5

57

5

8

5

58

307

9

5

59

11

10

23

60

5

11

5

61

31

12

5

62

5

13

5

63

125

14

5

64

5

15

17

65

5

16

5

66

23

17

121

67

5

18

5

68

67

19

5

69

5

20

11

70

541

21

5

71

17

22

53

72

5

23

5

73

41

24

5

74

5

25

11

75

185

26

5

76

11

27

31

77

5

28

5

78

25

29

347

79

5

30

5

80

95

31

5

81

5

32

23

82

1073

33

5

83

17

34

103

84

5

35

5

85

55

36

5

86

5

37

17

87

271

38

5

88

11

39

53

89

5

40

5

90

37

41

1133

91

5

42

11

92

131

43

5

93

11

44

31

94

3203

45

5

95

23

46

181

96

5

47

11

97

73

48

5

98

5

49

23

99

401

50

5

100

17

2.3.Envelope for more coordinates (1600
coordinates up to N=10250)

2.4.Envelopes for other
problem-configuration: 2S/3N, 2S/5N , 2S/11N

It was interesting,how
the envelopes of other configurations would behave. Surprise: they all have the
same slope in the trend. That means, the rate-of-approximation is much related.

Plot 3:
Envelope-curve for approximation 2S/3N, 2S/5N,
2S/11N

The longer line
for k=5
indicates, that this configuration approximates faster to zero

To see the
deviation from the trend (with slope –1) I rotate the whole plot by –45° :

Plot
4:Envelope-curves, rotated by 45°

The plot gives
the impression as if the curves could be separated using fourier-analysis; but
I don't have experience with this.

2.5.An earlier discussion in sci.math
research (may 2004)

I took only the
relevant posts, also shortened (or even removed) needless quoting of previous
post and formatted text to special style. Use the google-link to look at the
original posts.

Q = product(log Ak) and C depends only on n and
on d, the maximum degree of the ak.

In
particular,

let n=2 and d=1; let a1=2, a2=3; and
write p = b1, q = -b2.

Then this
says

F
= |p log 2 - q log 3| either
F = 0 or |F| > max(p,q)^-C

for a
constant C.

C can be found explicitly. Baker's book
doesn't give details of how, but they are out there in the literature. There
have been advances in this area since Baker's book. (Search Google for
"laurent mignotte nesterenko".) The value of C will still be much larger than
you'd like :-).

Anyway: if
the inequality above (the one you want to prove can't happen) holds then you
can find p,q
with |F| <=
2^-p or something like that. And if F is small then p and q have
to be somewhat similar in size.

Now, if p^-C < F <=
2^-p then -C log p <= -p log 2, so p / log p <= C
/ log 2. So you can get an upper bound on the size of p. It
may or may not be practical then to check all actual values of p up
to there.

A nasty
back-of-envelope calculation using a version of Laurent-Mignotte-Nestorenko
quoted in a paper I've found on the web suggests that actually you get some
such bound as

|F| >= exp(-17280) / p^(8640+1080 log p)

where those
numbers could doubtless be reduced by being less sloppy than I was, but
probably not hugely reduced.

So ... if I
haven't botched the above in any way (which I probably have), you
"only" need to check a few hundred thousand values of L.
You'd do that in practice by calculating log 3 / log 2 with sufficient accuracy (plain
ol' double precision should be fine) and looking to see how close to an
integer you can get by multiplying it by an integer up to a few hundred thousand
in size.

>
According to theorems mentioned at the start of chapter 3 of Baker's
"Transcendental Number Theory", the following is true.

..

> So
... if I haven't botched the above in any way (which

> I
probably have), you "only" need to check a few hundred

>
thousand values of L. You'd do that in practice by calculating

> log
3 / log 2 with sufficient accuracy (plain ol' double

>
precision should be fine) and looking to see how close to

> an
integer you can get by multiplying it by an integer

> up
to a few hundred thousand in size.

I just
received some e-mail from a friend of mine who knows much more about number theory than I do,
observing that (1) it's rather well
known (to those who know such things,
I suppose) that the 3n+1 conjecture itself follows from some sort of bounds on the approximability
of log 3 / log
2 by rationals, and (2)
that the required bounds are much
tighter than those obtainable by methods of the sort I mentioned. So if Gottfried Helms's conjecture
is strong enough to do much for the 3n+1
conjecture (I'm not sure what a
"primitive loop" is, I'm afraid) then the argument I sketched probably has some big holes in
it.

Contrariwise, if I've somehow managed to break my
perfect record of never posting anything to sci.math.research without at least one serious mistake, then
Gottfried's conjecture probably doesn't imply anything very exciting about
the Collatz conjecture. It's after midnight local time and I should be in
bed, so I shan't attempt to determine which of the three possibilities - the
third being that Gottfried and I are both right, and that Gottfried has found
a way to make progress on Collatz with weaker inequalities than others have
obtained -- is the truth. I'm just sounding a note of caution. :-)

where A is
the highest exponent keeping y integral (which is actually only a short form
for multiple steps of the
collatz-transformation, collecting all subsequent x/2 - operations) and allowing short form for recursive
notation:

z= T(y;B) = T(T(x;A);B) =
T(x;A,B)

then a loop of, for instance, length 2
occurs, if

z = T(x;A,B) = x

------------------------------------------

The occuring
equations are under investigation in some articles, that I've come across
(mostly via internet), and are obviously difficult to handle, but of high
general interest, as I learned this way.

My first
approach was to deal with any type of loop, the general form

x = T(x;A,B,C,D,..M)

= T(x;A,B,C,D,...,M,A,B,C,...M)

= T(x;A,B,C...)...

for what I've
got some nice results, but still not in the form of a general formulation,
which could, for instance, easily been transferred to 5x+1 and other classes
of the problem.

So I decided,
first to investigate a somehow "primitive" loop.

I assumed
most primitive loop (besides the trivial one, which is in this notation 1 = T(1;2) = T(1;2,2) = T(1;2,2,2,2,2,2,...) ) for my purposes the type of
loop, which starts with one or more ascending steps, and then descends in one
step. These are the assumed loops of the form

x = T(x;1,1,1,1,...,1,A)

where
immediately strong restrictions apply on A.

One reason
why I assumed that type as somehow primitive, is, that any eventual loop can
be expressed as a collection of ascending steps between descending steps, if
the length 0
is allowed.

So an
arbitrary transformation, for instance

y = T(x;1,1,3,1,4,2,1,1,3)

can be
segmented in

y = T( T(T(T(T(x;1,1,3);1,4);2);1,1,3)

and
eventually I can use my tools, that I developed in the analysis of the
"primitive" loop.

-------------------------------------------------------------

If I analyse
the transformation of length N

z =
T(x;1,1,1,1,...,1,A) where
the number of ones is denoted as "L",

and check,
whether it is ever possible, that we find a solution in integers, where z
equals x, I come to an expression of the right hand side of my inequality,
that I stated her in my previous postings, which also reflects an
*additional* restriction.

With an
arbitray 3-step-transformation, x' = T(x;A,B,C) with the length N=3, where x'
should equal x (thus realizing a loop) with x,y,z, the intermediate values of
each transformation, we can formulate a strong restriction on the exponents
A,B,C:

since y = (3x+1)/2^A , z = (3y+1)/2^B , x'=(3z+1)/2^C and x=x' assumed (to form a loop), we can write
their product as

Even more,
this equation shows without any lengthy proof, that whenever the sum S is
in general 2*N,
then *all* parentheses on the rhs must take their maximum, and this is 4 for
each of them.

This
restricts then all x,y,z to be 1 , which in turn restricts all A,B,C to
be 2.

Result:
if S =
2*N then x =
T(x;A,B,C,D...) only if A=B=C=D=...=2 and x = 1

----------------

It also shows
a widely unknown property of the loop-problem, that the values of x,y,z
have *high bounds* - which is important, since often articles about the
collatz-problem assume loops in "high number" areas, if they don't
find a solution in small accessible values. That is definitely not true: the
values of the members of an eventual loop are much restricted in values.

Since the trivial
loop is not of interest, we only have to study the cases

2^S < 4^N

and since 2^S
must be at least the next power of 2 after 3^N we have the inequality

powerceil2(3^N) <=
2^S < 4^N

Now from the
expansion of the three transformations into explicit formulas we get possible S; and I observed, that
they were *always* smaller than
powerceil2(3^N) in my cases of primitive loops. The contributions here
in sci.math.research , sci.math and by some private posts showed, that
there is an *extremely* high
likelihood, that this inequality never holds. However - that's still no proof, even not for this simple case.

----------------------------------------------------------

I think, I
made the needed progress now by investigating some modular classes, which
exhibited a useful, very simple structure for candidate numbers x and y.

The above
form of the inequality, stated for a "primitive" loop of length N with
L=N-1
ones and one exponent >1

x = T(x;1,1,1,1,1....1,C) with L ones, C>1 and x odd integer>1

This problem
can be separated into two:

y = T(x;1,1,1,1,1,1...1) with L=N-1 ones ("iterated
transformation")

z = T(y;C) where z should equal x

An iterated
transformation like y=T(x;1,1,1,1,1....1) with L ones restricts x and y to a
very specific modular structure. it comes out, that - with a free parameter i
ranging of nonnegative odd integers - x and y *must* have a very simple structure depending
on the length of the requested iterated transformation:

If this
inequality does not hold, then such a type of loop cannot exist -
irrespectively of any assumend length.

-------------------------------------------------------

I stated this
inequality for the case i=1 . For growing i the middle part of the
inequality is better written like the following

3^N - 1/i 2^N * --------- 2^N - 1/i

and this
converges to 3N,
which is definitevely smaller than the lhs:

for i->oo
the inequality

powerceil2(3^N) <= 3^N

is obviously
false.

So the case
with i=1
was the most critical case, and I accept the information and learned the arguments
for the extremely likelihood, that this inequality cannnot be satisfied for
any parameter L and C but which is still not *proven*....

------------------------------------------------------------------

If that's
solved, then at least a "primitive" loop can be actually negated -
which is surely no great step in the whole loop-problem or even the general
collatz problem - but... it's a start.

Therefore I'm
also more interested in extensible proofs than in min/max-estimations for separate
cases, if they cannot be generalized to
other collatz-type-problems.

...

Unfortunately,
a disproof of this type of collatz-loops is not conversely saying something
on the above inequality, insofar it concerns unproved assumtions about the
2-log of 3 and the like, what you and others had derived here - that's bad
luck. But I've got the impression, that some of my equations in this context
could be helpful for progress even in that regard. I'll post them, if I've
time these days.

Hope I
answered the questions, that you rose in your post, even if I had
difficulties to really follow your four segments today... :-)

> According to theorems mentioned at the
start of chapter 3 of Baker's "Transcendental Number Theory", the
following is true.
(...)

Hi Gareth -

that's a very
interesting material, thank you (hope, I'll ever learn to apply that myself).
I'll study it in more detail next days, since I'm guessing another connection
to that field here, where my derivations on number-structure possibly could
give additional information - or at least some more insight - I'll post it another
day.

For an
explanation for the origin of my question please see my other post.

Is this what
a "primitive cycle" actually is? A circuit is a cycle with one rise
and one fall.

In fact, B.
deWeger has recently shown that there is no cycle in the 3x+1 rproblem with less than 69
rises and falls in it. I can actually extend this to 70 and 71 rises and falls with Rhin's
inequality which I mentioned in an earlier post.

Hmm, that's
interesting. I called it a "primitive", if there are N-1
steps ascending and 1 step descending, like it is in (7->11->17)
-> 13 which, written as an transfer only noting the occuring
exponents-of-2

13 = T(7;1,1,4)

My question
was

"is there any solution in integers
x,N,A with x = T(x;1,1,1,1...1,A) with the 1 (N-1)-times occuring establishing
an N-Step-Loop

".

I found this
very tight relation to the 3^N - 2^N properties, thus my question here.

The case of
only one-raise-one-fall, if that means

x = T(x;1,A)

would then
just be a special case of my question and can be proven by enumeration or
even modular arithmetic.

But I would
like to know, how you accessed the problem of this specific of
rises&falls? I have a formula, how you can disprove a circle of *any*
finite combination of raises&falls in finite number of tests, let say 100
raises&falls by a number Z of tests, where Z is a combinatorical function of
about 42,
which I'm bounding down by optimizing my formulas.

My attempt
was to generalize these formulas to *unlimited* Z.

The
"primitive" Loop is the most simple structure of that general
problem of unlimited length, and has very tight and handy relations to the (3/2)^N
- structure and that of frac( (3/2)^N), which I'm currently investigating.

---------------------------

Concerning
your reference to deWegner:

His
literature looks interesting, especially those with the focus on binomials:
that was the next idea, that I wanted to step in. As I pointed out in a
previor post the related problem can be represented using a rep-unit-form and
the question, whether 3^n-1 div 2^N can ever be a repunit base a
certain power of 2^P with p reasonably greater than N (don't have it at hand just
now). I'm currently studying, what the binomial- expansion of 3^N = (2+1)^N
explains for that problem.

Just have
seen your reference to Rhin: could you point to a reference?